DOCSLIB.ORG

Speed of Sound in Water by a Direct Method 1 Martin Greenspan and Carroll E

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Speed of Sound in Water by a Direct Method 1 Martin Greenspan and Carroll E Journal of Rese arch of the National Bureau of Standards Vol. 59, No.4, October 1957 Research Paper 2795 Speed of Sound in Water by a Direct Method 1 Martin Greenspan and Carroll E. Tschiegg The speed of sound in distilled water wa,s m easured over the temperature ra nge 0° to 100° C with an accuracy of 1 part in 30,000. The results are given as a fifth-degree poly­ nomial and in tables. The water was contained in a cylindrical tank of fix ed length, termi­ nated at each end by a plane transducer, and the end-to-end time of flight of a pulse of sound was determined from a measurement of t he pulse-repetition frequency required to set the successive echoes into t im e coincidence. 1. Introduction as the two pulses have different shapes, the accurac.,' with which the coincidence could be set would be The speed of sound in water, c, is a physical very poor. Instead, the oscillator is run at about property of fundamental interest; it, together with half this frequency and the coincidence to be set is the density, determines the adiabatie compressibility, that among the first received pulses corresponding to and eventually the ratio of specific heats. The vari­ a particular electrical pulse, the first echo correspond­ ation with temperature is anomalous; water is the ing to the electrical pulse next preceding, and so on . only pure liquid for which it is known that the speed Figure 1 illustrates the uccessive signals correspond­ of sound does not decrcase monotonically with ing to three electrical input pul es. The input pulses temperature. fall halfway between the pulses for which the coinci­ There is also a practical interest in c in that water dence is set, so that they do not tend to overload is used as a standard liquid for the calibration of the amplifier or distort the oscill oscope traces. The instruments that measure the speed of sound in period of the oscillator, when properly se t, multi­ liquids automatically, both in the laboratory and in plied by twice the length of Ul e tank, is the speed of the field. In fact, it was in connection with the cali­ sound in the sample. bration of such "velocimeters" [1] 2 that our interest The oscilloscope trace actually looks like that in this work was first aroused. In the first place, shown in the in se t (fig. 1). The firs t cycle corre­ the available data scatter widely, as recent sum­ sponds to sound reHected from the inner faces only maries [2 , 3] clearly show. In many cases, the di s­ of the transducers, whereas the succeeding cycle crepancies far exceed the claimed accuracy or at correspond to sound reflected. one or more times from least the precision of the methods, even when the an outer face. Therefore, the coincidence is set by methods compared are the same. In the second maximizing the peak on eitllcr the first or second place, there exists no set of data that gives a smooth balf-cycle; the same result is obtained in either case variation with temperature over any considerable but the second half cycle is easier to use because it range. In particular, the best of these data yield is bigger. What we are mea uring h ere is the speed calibration curves for OUT velocimeters which are corresponding to the fi1" t arrival of the signal ; in a badly curved instead of straight (as they should be), nondispersive liquid this is the same as the phase and about which the data scatter irregularly, but velocity. It is true that the coin cidence is made at a reproducibly. The results here presented are free of these objections. 2. Method At the top of figure 1 is a schematic of the appa­ ratus. The sample is confined in a tube of which the ends are plane, parallel, electroacoustic trans­ ducers, quartz crystals in this case. If the left-hand crystal, say, is excited by a short pulse from the blocking oscillator, the oscilloscope, which measures the voltage on tne right-hand crystal, will show a received pulse and a series of echoes, as indicated n in idealized form on the line below (fig. 1). The pulse repetition frequency of the blocking oscillator is controlled by a sine-wave oscillator, and if this I frequency were adjusted so that each blocking oscil­ n lator pulse coincided with the first received pulse of the next preceding cycle, then the oscillator period I ~I would equal the time of flight of the pulse. However, n FIGURE 1. Schematic of method. I 1'bis work was supported in part by the Omce of Naval R esearch under contract NA-oTU"-7<J-4S. 'fhe three lower lines show in idealized form the events co rresponding to three , Figures in brackets indicate the literature referen ces at the end of this paper. sneccssh'e electri cal pulses. The short , thick line represents Lhe input pulse. 249 ' Lime one-fourth or three-quarters of the transducer period later Lhan the til!le o~ first arrival, by. which time there is opportumty for sOlll;d travclmg .by paths other than the shortest to affect the 10e~ t JOn of the maximum. However, the results are lllde­ pendent of whether til e first or second half c'y cl ~ is used · they are also not affected by substltutmg crystals of twice the thickness, or by changing the diameters of tlle tank, or of the hot electrodes. These results lead us to believe that the error introduced by this maximization technique is negligible. The question has been examined also in another way. Suppose a coin cidence to have been mac~e at frequency j; others can then be ma.de at submultlJ?les of j. At the frequency / /2 for lI1 stan?e, the. first received pulse correspondmg to a partIcular mput FIGUR E 2. Delay line, disassembled. pulse coincides with the second echo (not the first, Above tbe tank are tbe plugs whicb close the filling holes. and at the ends are tbe caps tbrougb which pass the electri cal cables and wblcb also clam p t he as before) corresponding to the electrical pulse next crystals seen in the foreg round. preceding, and so on. Effectively, the sOllnd pul~ e is timed over a path twice as long as before. It IS found that the measurements at j and ncar j /2 are substantially id entical, so that the error in question is less than, or at most comparable to, the expen­ mental error of the time meaS llrement. 3 . Appar a tus 3 .1. The Delay Line The disassembled delay line is shown in the photo­ graph, figure 2. The length of the tank i s about 200 mm, and the bore about 13 mm. The filling holes are sealed by plugs having T eflon gaskets; a small hole in one plug provides pressure.r elease. The tank is of a chromium steel 3 which, after heat treatmen t, takes a good optical fini sh. Because this steel is not so corrosion r esistant as the nickel­ chromium stainless steels, the bore of the tank was heavily gold plated. The ends of the tank are optically flat and parallel to within less than 1 f.1. To these ends are carefully wrung the 0.8-mm thick x-cut quartz crystals, which also are optically flat. The caps, when bolted on, clamp the crystals through neoprene O-rings. A FIGURE 3. Schematic oj one end of the tank, showing th e crystal coaxial cable passes through a seal in each cap, and and cap assembly. the center conductor makes contact with the outer 1. 'rank; 2. cap; 3. pl ug sealed with Teflon O-ring (thermocouple pasSf'S t hro ugh pressure relief tu be); 4. quartz crystal wrun g on to end of tank (springs (hot) electrode of the crystal through a light spring. make conta ct to electrodes) ; 5, neoprene O·ring; and 6, coaxial cable. The outer electrode is a 9 mm circle of aluminum­ backed pressure-sensitive adhesive tape. The inner was th e tank and heat-treated together with it. (ground) electrode is of fired-on gold and is about From these data, the length of the sound path is 12 mm in diameter. Contact is made through a known to better than 2 parts in 105 at any tempera­ light gold-plated helical spring which touches the ture between 0° and 100 0 C. It is, of course, electrode around the edge and bears on a shoulder necessary that the crystals be wrung down with machined into the bore. The inner electrodes and great care so that the fringes disappear all around springs are unnecessary if the sample has high con­ the periphery, to achieve this accuracy. The clamp­ ductivity or a high dielectric constant; they are ing gaskets must bear direc tly over the contacting usually omitted for water and aqueous solutions of surface and not spread out over the unsupported salts. Figure 3 is a schematic drawing of one end area, else the crystal will benel. With these pre­ of the assembly. cautions, the delay line may be disassembled and The length of the tank was measured at 20° C, reassembled repeatedly with reproducible results. and the coefficient of thermal expansion of the steel If the crystals have been properly wTung on and was measured on a sample cut from the same bar as clamped, they cannot be removed by hand after 3 F irtb-Sterling typelB-440A.
Load more

Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • Laplace and the Speed of Sound
    Laplace and the Speed of Sound By Bernard S. Finn * OR A CENTURY and a quarter after Isaac Newton initially posed the problem in the Principia, there was a very apparent discrepancy of almost 20 per cent between theoretical and experimental values of the speed of sound. To remedy such an intolerable situation, some, like New- ton, optimistically framed additional hypotheses to make up the difference; others, like J. L. Lagrange, pessimistically confessed the inability of con- temporary science to produce a reasonable explanation. A study of the development of various solutions to this problem provides some interesting insights into the history of science. This is especially true in the case of Pierre Simon, Marquis de Laplace, who got qualitatively to the nub of the matter immediately, but whose quantitative explanation performed some rather spectacular gyrations over the course of two decades and rested at times on both theoretical and experimental grounds which would later be called incorrect. Estimates of the speed of sound based on direct observation existed well before the Newtonian calculation. Francis Bacon suggested that one man stand in a tower and signal with a bell and a light. His companion, some distance away, would observe the time lapse between the two signals, and the speed of sound could be calculated.' We are probably safe in assuming that Bacon never carried out his own experiment. Marin Mersenne, and later Joshua Walker and Newton, obtained respectable results by deter- mining how far they had to stand from a wall in order to obtain an echo in a second or half second of time.
    [Show full text]
  • Introduction to Physical Chemistry – Lecture 5 Supplement: Derivation of the Speed of Sound in Air
    Introduction to Physical Chemistry – Lecture 5 Supplement: Derivation of the Speed of Sound in Air I. LECTURE OVERVIEW We can combine the results of Lecture 5 with some basic techniques in fluid mechanics to derive the speed of sound in air. For the purposes of the derivation, we will assume that air is an ideal gas. II. DERIVATION OF THE SPEED OF SOUND IN AN IDEAL GAS Consider a sound wave that is produced in an ideal gas, say air. This sound may be produced by a variety of methods (clapping, explosions, speech, etc.). The central point to note is that the sound wave is defined by a local compression and then expansion of the gas as the wave FIG. 2: An imaginary cross-sectional tube running from one passes by. A sound wave has a well-defined velocity v, side of the sound wave to the other. whose value as a function of various properties of the gas (P , T , etc.) we wish to determine. Imagine that our cross-sectional area is the opening of So, consider a sound wave travelling with velocity v, a tube that exits behind the sound wave, where the air as illustrated in Figure 1. From the perspective of the density is ρ + dρ, and velocity is v + dv. Since there is sound wave, the sound wave is still, and the air ahead of no mass accumulation inside the tube, then applying the it is travelling with velocity v. We assume that the air principle of conservation of mass we have, has temperature T and P , and that, as it passes through the sound wave, its velocity changes to v + dv, and its ρAv = (ρ + dρ)A(v + dv) ⇒ temperature and pressure change slightly as well, to T + ρv = ρv + vdρ + ρdv + dvdρ ⇒ dT and P + dP , respectively (see Figure 1).
    [Show full text]
  • The Physics of Sound 1
    The Physics of Sound 1 The Physics of Sound Sound lies at the very center of speech communication. A sound wave is both the end product of the speech production mechanism and the primary source of raw material used by the listener to recover the speaker's message. Because of the central role played by sound in speech communication, it is important to have a good understanding of how sound is produced, modified, and measured. The purpose of this chapter will be to review some basic principles underlying the physics of sound, with a particular focus on two ideas that play an especially important role in both speech and hearing: the concept of the spectrum and acoustic filtering. The speech production mechanism is a kind of assembly line that operates by generating some relatively simple sounds consisting of various combinations of buzzes, hisses, and pops, and then filtering those sounds by making a number of fine adjustments to the tongue, lips, jaw, soft palate, and other articulators. We will also see that a crucial step at the receiving end occurs when the ear breaks this complex sound into its individual frequency components in much the same way that a prism breaks white light into components of different optical frequencies. Before getting into these ideas it is first necessary to cover the basic principles of vibration and sound propagation. Sound and Vibration A sound wave is an air pressure disturbance that results from vibration. The vibration can come from a tuning fork, a guitar string, the column of air in an organ pipe, the head (or rim) of a snare drum, steam escaping from a radiator, the reed on a clarinet, the diaphragm of a loudspeaker, the vocal cords, or virtually anything that vibrates in a frequency range that is audible to a listener (roughly 20 to 20,000 cycles per second for humans).
    [Show full text]
  • FORMULAS for CALCULATING the SPEED of SOUND Revision G
    FORMULAS FOR CALCULATING THE SPEED OF SOUND Revision G By Tom Irvine Email: [email protected] July 13, 2000 Introduction A sound wave is a longitudinal wave, which alternately pushes and pulls the material through which it propagates. The amplitude disturbance is thus parallel to the direction of propagation. Sound waves can propagate through the air, water, Earth, wood, metal rods, stretched strings, and any other physical substance. The purpose of this tutorial is to give formulas for calculating the speed of sound. Separate formulas are derived for a gas, liquid, and solid. General Formula for Fluids and Gases The speed of sound c is given by B c = (1) r o where B is the adiabatic bulk modulus, ro is the equilibrium mass density. Equation (1) is taken from equation (5.13) in Reference 1. The characteristics of the substance determine the appropriate formula for the bulk modulus. Gas or Fluid The bulk modulus is essentially a measure of stress divided by strain. The adiabatic bulk modulus B is defined in terms of hydrostatic pressure P and volume V as DP B = (2) - DV / V Equation (2) is taken from Table 2.1 in Reference 2. 1 An adiabatic process is one in which no energy transfer as heat occurs across the boundaries of the system. An alternate adiabatic bulk modulus equation is given in equation (5.5) in Reference 1. æ ¶P ö B = ro ç ÷ (3) è ¶r ø r o Note that æ ¶P ö P ç ÷ = g (4) è ¶r ø r where g is the ratio of specific heats.
    [Show full text]
  • Speed of Sound in a System Approaching Thermodynamic Equilibrium
    Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 61 (2016) 842 Speed of Sound in a System Approaching Thermodynamic Equilibrium Arvind Khuntia1, Pragati Sahoo1, Prakhar Garg1, Raghunath Sahoo1∗, and Jean Cleymans2 1Discipline of Physics, School of Basic Science, Indian Institute of Technology Indore, Khandwa Road, Simrol, M.P. 453552, India 2UCT-CERN Research Centre and Department of Physics, University of Cape Town, Rondebosch 7701, South Africa Introduction uses Experimental high energy collisions at 1 fT (E) ≡ : (1) RHIC and LHC give an opportunity to study E−µ the space-time evolution of the created hot expq T ± 1 and dense matter known as QGP at high ini- tial energy density and temperature. As the where the function expq(x) is defined as initial pressure is very high, the system un- ( dergo expansion with decreasing temperature [1 + (q − 1)x]1=(q−1) if x > 0 exp (x) ≡ and energy density till the occurrence of the fi- q [1 + (1 − q)x]1=(1−q) if x ≤ 0 nal kinetic freeze-out. This change in pressure with energy density is related to the speed of (2) sound inside the system. The QGP formed and, in the limit where q ! 1 it re- in heavy ion collisions evolves from the initial duces to the standard exponential; QGP phase to a hadronic phase via a possi- limq!1 expq(x) ! exp(x). In the present con- ble mixed phase. The speed of sound reduces text we have taken µ = 0, therefore x ≡ E=T to zero on the phase boundary in a first or- is always positive.
    [Show full text]
  • Acoustics: the Study of Sound Waves
    Acoustics: the study of sound waves Sound is the phenomenon we experience when our ears are excited by vibrations in the gas that surrounds us. As an object vibrates, it sets the surrounding air in motion, sending alternating waves of compression and rarefaction radiating outward from the object. Sound information is transmitted by the amplitude and frequency of the vibrations, where amplitude is experienced as loudness and frequency as pitch. The familiar movement of an instrument string is a transverse wave, where the movement is perpendicular to the direction of travel (See Figure 1). Sound waves are longitudinal waves of compression and rarefaction in which the air molecules move back and forth parallel to the direction of wave travel centered on an average position, resulting in no net movement of the molecules. When these waves strike another object, they cause that object to vibrate by exerting a force on them. Examples of transverse waves: vibrating strings water surface waves electromagnetic waves seismic S waves Examples of longitudinal waves: waves in springs sound waves tsunami waves seismic P waves Figure 1: Transverse and longitudinal waves The forces that alternatively compress and stretch the spring are similar to the forces that propagate through the air as gas molecules bounce together. (Springs are even used to simulate reverberation, particularly in guitar amplifiers.) Air molecules are in constant motion as a result of the thermal energy we think of as heat. (Room temperature is hundreds of degrees above absolute zero, the temperature at which all motion stops.) At rest, there is an average distance between molecules although they are all actively bouncing off each other.
    [Show full text]
  • James Clerk Maxwell and the Physics of Sound
    James Clerk Maxwell and the Physics of Sound Philip L. Marston The 19th century innovator of electromagnetic theory and gas kinetic theory was more involved in acoustics than is often assumed. Postal: Physics and Astronomy Department Washington State University Introduction Pullman, Washington 99164-2814 The International Year of Light in 2015 served in part to commemorate James USA Clerk Maxwell’s mathematical formulation of the electromagnetic wave theory Email: of light published in 1865 (Marston, 2016). Maxwell, however, is also remem- [email protected] bered for a wide range of other contributions to physics and engineering includ- ing, though not limited to, areas such as the kinetic theory of gases, the theory of color perception, thermodynamic relations, Maxwell’s “demon” (associated with the mathematical theory of information), photoelasticity, elastic stress functions and reciprocity theorems, and electrical standards and measurement methods (Flood et al., 2014). Consequently, any involvement of Maxwell in acoustics may appear to be unworthy of consideration. This survey is offered to help overturn that perspective. For the present author, the idea of examining Maxwell’s involvement in acous- tics arose when reading a review concerned with the propagation of sound waves in gases at low pressures (Greenspan, 1965). Writing at a time when he served as an Acoustical Society of America (ASA) officer, Greenspan was well aware of the importance of the fully developed kinetic theory of gases for understanding sound propagation in low-pressure gases; the average time between the collision of gas molecules introduces a timescale relevant to high-frequency propagation. However, Greenspan went out of his way to mention an addendum at the end of an obscure paper communicated by Maxwell (Preston, 1877).
    [Show full text]
  • The Speed of Sound
    SOL 5.2 PART 3 Force, Motion, Energy & Matter NOTEPAGE FOR STUDENT The Speed of Sound We know that sound is a form of energy that is produced by vibrating matter. We also know that where there is no matter there is no sound. Sound waves must have a medium (matter) to travel through. When we talk, sound waves travel in air. Sound also travels in liquids and solids. How fast speed travels, or the speed of sound, depends on the kind of matter it is moving through. Of the three phases of matter (gas, liquid, and solid), sound waves travel the slowest through gases, faster through liquids, and fastest through solids. Let’s find out why. Sound moves slowest through a gas. That’s because the molecules in a gas are spaced very far apart. In order for sound to travel through air, the floating molecules of matter must vibrate and collide to form compression waves. Because the molecules of matter in a gas are spaced far apart, sound moves slowest through a gas. Sound travels faster in liquids than in gases because molecules are packed more closely together. This means that when the water molecules begin to vibrate, they quickly begin to collide with each other forming a rapidly moving compression wave. Sound travels over four times faster than in air! Sound travels fastest through solids. This is because molecules in a solid are packed against each other. When a vibration begins, the molecules of a solid immediately collide and the compression wave travels rapidly. How fast, you ask? Sound waves travel over 17 times faster through steel than through air.
    [Show full text]
  • Echo-Based Measurement of the Speed of Sound Alivia Berg and Michael Courtney U.S
    Echo-based measurement of the speed of sound Alivia Berg and Michael Courtney U.S. Air Force Academy, 2354 Fairchild Drive, USAF Academy, CO 80840 [email protected] Abstract The speed of sound in air can be measured by popping a balloon next to a microphone a measured distance away from a large flat wall, digitizing the sound waveform, and measuring the time between the sound of popping and return of the echo to the microphone. The round trip distance divided by the time is the speed of sound. Introduction Echo technology and time based speed of sound measurements are not new,[1-3] but this technique allows a student lab group or teacher performing a demonstration to make precise speed of sound measurements with a notebook computer, Vernier LabQuest, or other sound digitizer that allows the sound pressure vs. time waveform to be viewed. Method The method employed here used the sound recording and editing program called Audacity.[4] The program was used to digitize, record, and display the sound waveform of a balloon being popped next to a microphone a carefully measured distance (45.72 m) from a large, flat wall (side of a building) from which the sound echos. The sample rate was 100000 samples per second. Three balloons were tested. The weather was sunny with a dew point of 16.1°C and temperature of 18.3°C. The elevation at the site was 360 m. Times of popping and echo return were determined by visual inspection of the sound waveform, zooming in and usng the program cursor as necessary, and then subtracting the pop time from the echo return time to determine the round trip sound travel time.
    [Show full text]
  • Speed of Sound - Temperature Matters, Not Air Pressure Die Schallgeschwindigkeit, Die Temperatur
    Speed of sound - temperature matters, not air pressure Die Schallgeschwindigkeit, die Temperatur ... und nicht der Luftdruck http://www.sengpielaudio.com/DieSchallgeschwindigkeitLuftdruck.pdf The expression for the speed of sound c0 in air is: p c = 0 Eq. 1 0 UdK Berlin Sengpiel c0 = speed of sound in air at 0°C = 331 m/s Speed of sound at 20°C (68°F) is c20 = 343 m/s 12.96 3 Tutorium p0 = atmospheric air pressure 101,325 Pa (standard) Specific acoustic impedance at 20°C is Z20 = 413 N · s /m 3 3 0 = density of air at 0°C: 1.293 kg/m = Z0 / c0 Air density at 20°C is 20 = 1.204 kg/m 3 = lower-case Greek letter "rho" Specific acoustic impedance at 0°C is Z0 = 428 N · s /m = adiabatic index of air at 0°C: 1.402 = cp / cv = ratio of the specific warmth = lower-case Greek letter "kappa" Sound resistance = Specific acoustic impedance Z = · c Calculating from these values, the speed of sound c0 in air at 0°C: p 101325 0 The last accepted measure shows the value 331.3 m/s. c0 1,402 331.4 m /s Eq. 2 0 1,2935 And from this one gets the speed of sound: c = c0 · 1 in m/s Eq. 3 = coefficient of expansion 1 / 273.15 = 3.661 · 10–3 in 1 / °C –273.15°C (Celsius) = absolute zero = 0 K (Kelvin), all molecules are motionless. = temperature in °C = lower-case Greek letter "theta" c0 = speed of sound in air at 0°C The speed of sound c20 in air of 20°C is: c20 = 343 m/s This value of c is usually used in formulas.
    [Show full text]
  • Speed of Sound
    An Internet Book on Fluid Dynamics Speed of Sound The next step in our treatment of compressible flows is to develop relations for and understanding of the speed of sound in a compressible fluid. Sound consists of small amplitude pressure waves propagating through the fluid and, in most of the analysis which follows we will retain only the mathematical terms that are linear in the pressure perturbations while neglecting all terms that are quadratic or higher order in those perturbations. We begin with the following thought experiment. Consider a rigid tube of cross-sectional area, A, that contains a piston that is initially at rest. The tube is filled with gas at rest with pressure, p, temperature, T , and density, ρ, but at time t = 0 it is suddenly given a small velocity, Δu, directed into the gas as sketched in Figure 1. This generates a wave that travels down the pipe with velocity, V ,and Figure 1: Tube with piston set in motion at t = 0, with small velocity, Δu. leaves behind it gas that now has a velocity, Δu, and increased pressure, p +Δp, temperature, T +ΔT , and density, ρ +Δρ. We seek to determine the wave velocity, V , by applying the basic conservation laws Figure 2: Flow with wave at rest. that apply to the gas flow. To do so we examine the flow in a frame fixed in the wave as sketched in Figure 2 by applying a Galilean transformation to the flow in Figure 1. Conservation of mass across the wave then requires that A(ρ +Δρ)(Δu − V )=−ρV A (Bod1) so that V Δρ Δu = (Bod2) (ρ +Δρ) In addition, applying the momentum equation to the wave yields AΔp = AρV 2 − A(ρ +Δρ)(Δu − V )2 (Bod3) that, using the continuity result (Bod2), leads to Δp Δρ V 2 = 1+ (Bod4) Δρ ρ The speed of sound is defined as the speed of the wave for very small perturbations or V → c when Δu → 0(andsoΔp, ΔT,Δρ → 0) (Bod5) and therefore dp c2 = (Bod6) dρ This result holds for any substance whether solid, liquid or gas and, at various points throughout this book we will use it for many different substances.
    [Show full text]